On Legendrian knots and polynomial invariants

It is proved in this note that the analogues of the Bennequin inequality which provide an upper bound for the Bennequin invariant of a Legendrian knot in the standard contact three dimensional space in terms of the least degree in the framing variable of the HOMFLY and the Kauffman polynomials are not sharp. Furthermore, the relationships between these restrictions on the range of the Bennequin invariant are investigated, which leads to a new simple proof of the inequality involving the Kauffman polynomial.

     

Chern–Simons Invariants of Torus Links

We compute the vacuum expectation values of torus knot operators in Chern–Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus knots and links, and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.     

Delta move and polynomial invariants of links

We investigate how a self-delta move, which is a delta move on the same component, influences the HOMFLY polynomial of a link. Then we reveal some relationships among finite type invariants, which are coming from the derivatives of the Jones polynomials and the first HOMFLY coefficient polynomials, of the four links involving in a self-delta move.     

q-holonomic formulas for colored HOMFLY polynomials of 2-bridge links

We compute q-holonomic formulas for the HOMFLY polynomials of 2-bridge links colored with one-column (or one-row) Young diagrams.     

Unsigned State Models for the Jones Polynomial

It is a well-known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the process of deriving this result, we show that for every diagram of a link in S ³ there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating to the Jones and Bollobás-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.     

Cabling procedure for the colored HOMFLY polynomials

We discuss using the cabling procedure to calculate colored HOMFLY polynomials. We describe how it can be used and how the projectors and $\mathcal{R}$ -matrices needed for this procedure can be found. The constructed matrix expressions for the projectors and $\mathcal{R}$ -matrices in the fundamental representation allow calculating the HOMFLY polynomial in an arbitrary representation for an arbitrary knot. The computational algorithm can be used for the knots and links with ¦Q¦m ≤ 12, where m is the number of strands in a braid representation of the knot and ¦Q¦ is the number of boxes in the Young diagram of the representation. We also discuss the justification of the cabling procedure from the group theory standpoint, deriving expressions for the fundamental $\mathcal{R}$ -matrices and clarifying some conjectures formulated in previous papers.     

Links in a thickened surface

For a given two-dimensional surface μ, we studi invariants for oriented links in μ×[0,1] which generalize the two-variable HOMFLY polynomials when μ is the 2-disk. These invariants are connected to multiparameter quantum groups whose special properties are discussed. Si studiano gli invarianti dei nodi in spazi del tipo μ×[0,1], dove μ è una superficie. Questi invarianti generalizzano gli invarianti di HOMFLY.

---

(Conferenza tenuta dal Prof. P. Cotta-Ramusino il 16 maggio 1991)     

Maximal Bennequin numbers and Kauffman polynomials of positive links

By using results of Yamada and of Yokota, concerning link diagrams and link polynomials, we give some relationships between maximal Bennequin numbers and Kauffman polynomials of positive links.

     

Genus expansion of HOMFLY polynomials

In the planar limit of the’ t Hooft expansion, the Wilson-loop vacuum average in the three-dimensional Chern-Simons theory (in other words, the HOMFLY polynomial) depends very simply on the representation (Young diagram), H_R(A|q)|_q=1 = (σ₁(A)^|R|. As a result, the (knot-dependent) Ooguri-Vafa partition function $\sum\nolimits_R {H_{R\chi R} \left\{ {\bar pk} \right\}}$ becomes a trivial τ -function of the Kadomtsev-Petviashvili hierarchy. We study higher-genus corrections to this formula for H_R in the form of an expansion in powers of z = q ? q^?1. The expansion coefficients are expressed in terms of the eigenvalues of cut-and-join operators, i.e., symmetric group characters. Moreover, the z-expansion is naturally written in a product form. The representation in terms of cut-and-join operators relates to the Hurwitz theory and its sophisticated integrability. The obtained relations describe the form of the genus expansion for the HOMFLY polynomials, which for the corresponding matrix model is usually given using Virasoro-like constraints and the topological recursion. The genus expansion differs from the better-studied weak-coupling expansion at a finite number N of colors, which is described in terms of Vassiliev invariants and the Kontsevich integral.     

Some families of links with divergent Mahler measure

In the following note we develop a method to prove that the Mahler Measure of the Jones polynomial of a family of links diverges. We apply this to several examples from the literature. We then use the W-polynomial to find the Kauffman Bracket of some families of Montesinos links and show that their Jones polynomials too have divergent Mahler measure.     

Kauffman polynomials of some links and invariants of 3-manifolds

Kauffman bracket polynomials of the so-called generalized tree-like links are studied. An algorithm of Witten type invariants, which was defined by Blanchet and Habegger et al. of more general 3-manifolds is given.     

On the Nontriviality of the Polynomial Invariant of Two-Component Link

ＯｎｔｈｅＮｏｎｔｒｉｖｉａｌｉｔｙｏｆｔｈｅＰｏｌｙｎｏｍｉａｌＩｎｖａｒｉａｎｔｏｆＴｗｏ－ＣｏｍｐｏｎｅｎｔＬｉｎｋ￥ＷａｎｇＸｉｎｇｙｕ；ＺｈａｏＧｕａｎｇｆｅｎｇ（ＨｅｎａｎＵｎｉｖｅｒｓｉｔｙ，Ｋａｉｆｅｎｇ，Ｈｅｎａｎ，４７５００１）（...     

Skein modules and the noncommutative torus

We prove that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus. The proof is based on Chebyshev polynomials. As an application, we describe the structure of the Kauffman bracket skein module of a solid torus as a module over the algebra of the cylinder over a torus, and recover a result of Hoste and Przytycki about the skein module of a lens space. We establish simple formulas for Jones-Wenzl idempotents in the skein algebra of a cylinder over a torus, and give a straightforward computation of the -th colored Kauffman bracket of a torus knot, evaluated in the plane or in an annulus.

     

On the relation between the A-polynomial and the Jones polynomial

This paper shows that the noncommutative generalization of the A-polynomial of a knot, defined using Kauffman bracket skein modules, together with finitely many colored Jones polynomials, determines the remaining colored Jones polynomials of the knot. It also shows that under certain conditions, satisfied for example by the unknot and the trefoil knot, the noncommutative generalization of the A-polynomial determines all colored Jones polynomials of the knot.

     

Challenges of β-deformation

We briefly review problems arising in the study of the beta deformation, which turns out to be the most difficult element in a number of modern problems: the deviation of ?? from unity is connected with the ??exit from the free-fermion point?? in two-dimensional conformal theories, from the symmetric graviphoton field with ??₂ = ???₁ in instanton sums in four-dimensional supersymmetric Yang-Mills theories, with the transition from matrix models to beta ensembles, from HOMFLY polynomials to superpolynomials in the Chern-Simons theory, from quantum groups to elliptic and hyperbolic algebras, and so on. We mainly attend to issues related to the Alday-Gaiotto-Tachikawa correspondence and its possible generalizations.     

Are there <Emphasis Type="Italic">p</Emphasis>-adic knot invariants?

We suggest using the Hall–Littlewood version of the Rosso–Jones formula to define the germs of p-adic HOMFLY-PT polynomials for torus knots [m, n] as coefficients of superpolynomials in a q-expansion. In this form, they have at least the [m, n] ? [n, m] topological invariance. This opens a new possibility to interpret superpolynomials as p-adic deformations of HOMFLY polynomials and poses a question of generalizing to other knot families, which is a substantial problem for several branches of modern theory.     

Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams

We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries, these numbers of matrices are not polynomials in q (Stembridge in Ann. Comb. 2(4):365, 1998); however, when the set of entries is a Young diagram, the numbers, up to a power of q?1, are polynomials with nonnegative coefficients (Haglund in Adv. Appl. Math. 20(4):450, 1998). In this paper, we give a number of conditions under which these numbers are polynomials in q, or even polynomials with nonnegative integer coefficients. We extend Haglund’s result to complements of skew Young diagrams, and we apply this result to the case where the set of entries is the Rothe diagram of a permutation. In particular, we give a necessary and sufficient condition on the permutation for its Rothe diagram to be the complement of a skew Young diagram up to rearrangement of rows and columns. We end by giving conjectures connecting invertible matrices whose support avoids a Rothe diagram and Poincaré polynomials of the strong Bruhat order.     

Skeins associated with Homfly and Kauffman polynomials and invariants of graphs

In this article, we treat skeins associated with Homfly and Kauffman polynomials. Several properties of skeins are studied, and as an application, we construct invariants of trivalent graphs in S³.     

On the colored Tutte polynomial of a graph of bounded treewidth

We observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601-622] for the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak [An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39-54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005) 276-290], using a different algorithm based on logical techniques.     

Rectangular superpolynomials for the figure-eight knot 4<Subscript>1</Subscript>

We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot 4₁ in an arbitrary rectangular representation R = [r^s] as a sum over all Young subdiagrams λ of R with surprisingly simple coefficients of the Z factors. Intriguingly, these coefficients are constructed from the quantum dimensions of symmetric representations of the groups SL(r) and SL(s) and restrict the summation to diagrams with no more than s rows and r columns. Moreover, the β-deformation to Macdonald dimensions yields polynomials with positive integer coefficients, which are plausible candidates for the role of superpolynomials for rectangular representations. Both the polynomiality and the positivity of the coefficients are nonobvious, nevertheless true. This generalizes the previously known formulas for symmetric representations to arbitrary rectangular representations. The differential expansion allows introducing additional gradings. For the trefoil knot 3₁, to which our results for the knot 4₁ are immediately extended, we obtain the so-called fourth grading of hyperpolynomials. The property of factorization in roots of unity is preserved even in the five-graded case.